Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{ll}1 & -3 \\ 1 & -2\end{array}\right]\) and \(\left[\begin{array}{ll}-2 & 3 \\ -1 & 1\end{array}\right]\)

Step-by-Step Solution

Answer

We will multiply the given matrices \(\textbf{A}\) and \(\textbf{B}\) and check if the result is an identity matrix: \(\textbf{A} \cdot \textbf{B} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right] \cdot \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]\) Calculating the elements of the resulting matrix \(\textbf{C}\), we get: \(\textbf{C} = \left[\begin{array}{ll}1 & 0 \\\ 0 & 1\end{array}\right]\) Since the product of \(\textbf{A}\) and \(\textbf{B}\) results in an identity matrix, the matrices \(\textbf{A}\) and \(\textbf{B}\) are inverses of each other.

Step 1: Multiply the two matrices together

First, let's multiply the given matrices \(\textbf{A}\) and \(\textbf{B}\) together: \(\textbf{A} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right]\), \(\textbf{B} = \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]\) \(\textbf{C} = \textbf{A} \cdot \textbf{B} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right] \cdot \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]\)

Step 2: Calculate the elements of the resulting matrix

To find the elements of the resulting matrix \(\textbf{C}\), let's use matrix multiplication rules. Multiply the elements of each row in \(\textbf{A}\) by the corresponding elements of each column in \(\textbf{B}\), and then add the results: \(c_{11} = (1 \times -2) + (-3 \times -1) = -2 + 3 = 1\) \(c_{12} = (1 \times 3) + (-3 \times 1) = 3 - 3 = 0\) \(c_{21} = (1 \times -2) + (-2 \times -1) = -2 + 2 = 0\) \(c_{22} = (1 \times 3) + (-2 \times 1) = 3 - 2 = 1\)

Step 3: Write the resulting matrix

Now that we have calculated all the elements of matrix \(\textbf{C}\), let's write them as a matrix: \(\textbf{C} = \left[\begin{array}{ll}1 & 0 \\\ 0 & 1\end{array}\right]\)

Step 4: Compare the matrices

Since the product of \(\textbf{A}\) and \(\textbf{B}\) resulted in an identity matrix, we can conclude that the matrices \(\textbf{A}\) and \(\textbf{B}\) are inverses of each other.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. It involves multiplying rows of one matrix by columns of another. It's crucial to understand the rules to correctly perform this operation.For two matrices to be multipliable, the number of columns in the first matrix must equal the number of rows in the second matrix. Each element in the resulting product matrix is calculated by taking the dot product of each row of the first matrix with each column of the second matrix.Here's how it works:

• Multiply corresponding elements of each row and column pair.
• Add those products together to get a single entry in the resulting matrix.

In our exercise, we demonstrated this by multiplying matrix \(\textbf{A} = \left[\begin{array}{ll}1 & -3 \ 1 & -2\end{array}\right]\) and \(\textbf{B} = \left[\begin{array}{ll}-2 & 3 \ -1 & 1\end{array}\right]\).Always remember to carefully follow the row by column multiplication method to ensure accurate matrix multiplication.

Identity Matrix

The identity matrix is an essential concept in matrix algebra. It acts like the number 1 in matrix multiplication. When any matrix is multiplied by its identity matrix, the original matrix is returned.The identity matrix \(\textbf{I}_n\) for an \(n \times n\) matrix has ones on its main diagonal (from top-left to bottom-right) and zeroes elsewhere. For example, a 2x2 identity matrix looks like:\[\textbf{I}_2 = \left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]\]Here are key properties:

• Multiplying any \(n \times n\) matrix by \(\textbf{I}_n\) leaves the original matrix unchanged.
• The identity matrix serves as the matrix equivalent of multiplying by 1.

By confirming that the product of matrices \(\textbf{A}\) and \(\textbf{B}\) results in an identity matrix, as seen in our exercise, we demonstrate that they are indeed inverses of each other.

Matrix Algebra

Matrix algebra is a broad area of mathematics that deals with the manipulation and understanding of matrices. It includes operations such as addition, subtraction, and notably, multiplication like what we performed in this exercise.In matrix algebra, understanding the operations and properties of matrices is crucial. Here are some important aspects:

• Addition and Subtraction: Matrices of the same dimensions can be added or subtracted by adding or subtracting their corresponding elements.
• Multiplication: As we discussed, this involves rows and columns and requires careful adherence to multiplication rules.
• Inverses: A matrix \(\textbf{A}\) has an inverse (denoted \(\textbf{A}^{-1}\)) if multiplying \(\textbf{A}\) by \(\textbf{A}^{-1}\) results in the identity matrix.

Understanding these concepts allows us to solve system equations and perform transformations in various fields like computer graphics and data analysis. Being able to identify and compute critical matrices, like inverses, enhances proficiency in matrix algebra and is essential for solving complex mathematical problems.